To find the domain of the function \( f(x) = \frac{4x + 1}{x^2 + x - 6} \), we need to determine the values of \( x \) that make the function undefined. Since a rational function is undefined when its denominator equals zero, we begin by solving for the values of \( x \) that make the denominator zero.

### Step 1: Set the denominator equal to zero:
\[
x^2 + x - 6 = 0
\]

### Step 2: Factor the quadratic equation:
\[
x^2 + x - 6 = (x - 2)(x + 3) = 0
\]

### Step 3: Solve for \( x \):
\[
x - 2 = 0 \quad \text{or} \quad x + 3 = 0
\]
\[
x = 2 \quad \text{or} \quad x = -3
\]

### Step 4: Exclude these values from the domain:
Since the function is undefined when \( x = 2 \) and \( x = -3 \), the domain of the function is all real numbers except these two points.

### Final Answer:
The domain of \( f(x) \) is:
\[
\boxed{(-\infty, -3) \cup (-3, 2) \cup (2, \infty)}
\]

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:
1. How would the domain change if the numerator had a square root?
2. What are the vertical asymptotes of the function \( f(x) = \frac{4x + 1}{x^2 + x - 6} \)?
3. How do you find the horizontal asymptote of a rational function?
4. What would the graph of this function look like, and where would it have discontinuities?
5. Can you determine the behavior of the function near \( x = -3 \) and \( x = 2 \)?

**Tip:** When factoring quadratics, always check if the coefficients suggest a simple factoring pattern (like two binomials).

Explore how to find the domain of the rational function f(x) = (4x + 1) / (x^2 + x - 6). This detailed solution covers the factoring of quadratic equations and the exclusion of values where the function is undefined. Suitable for students in Grades 9-12.

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